# Core Concept

popsynth core function is to create observed surveys from latent population models.

First, let’s define what a population of objects is in terms of a generative model. The two main ingredients are the objects’ spatial distribution ($$\lambda(\vec{r}; \vec{\psi})$$) and the distribution of their inherent properties ($$\pi(\vec{\phi} | \vec{\psi})$$). Here, $$\vec{\psi}$$ are the latent population parameters, $$\vec{r}$$ are the spatial locations of the objects, and $$\vec{\phi}$$ are the properties of the individual objects (luminosity, spin, viewing angle, mass, etc.). The spatial distribution is defined such that:

$\frac{d \Lambda}{dt}(\vec{\psi}) = \int d r \frac{dV}{dr} \lambda(\vec{r}; \vec{\psi})$

is the intensity of objects for a given set of population parameters. With these definitions we can define the probability for an object to have position $$\vec{r}$$ and properties $$\vec{\phi}$$ as

$\pi(\vec{r}, \vec{\phi} | \vec{\psi}) = \frac{\lambda(\vec{r}; \vec{\psi}) \pi(\vec{\phi} | \vec{\psi})}{ \int d r \frac{dV}{dr} \lambda(\vec{r}; \vec{\psi})}$

popsynth allows you to specify these spatial and property distributions in an object-oriented way to create surveys. The final ingredient to creating a sample for a survey is knowing how many objects to sample from the population (before any selection effects are applied). Often, we see this number in simulation frameworks presented as “we draw N objects to guarantee we have enough.” This is incorrect. A survey takes place over a given period of time ($$\Delta t$$) in which observed objects are counted. This is a description of a Poisson process. Thus, the number of objects in a simulation of this survey is a draw from a Poisson distribution:

$N \sim \mathrm{Poisson}\left(\Delta t \frac{d\Lambda}{dt}\right) \mathrm{.}$

Thus, popsynth first numerically integrates the spatial distribution to determine the Poisson rate parameter for the given $$\vec{\psi}$$, then makes a Poisson draw for the number of objects in the population survey. For each object, positions and properties are drawn with arbitrary dependencies between them. Finally, selection functions are applied to either latent or observed (with or without measurement error) properties.

Note: If instead we draw a preset number of objects, as is done in many astrophysical population simulation frameworks, it is equivalent to running a survey up until that specific number of objects is detected. This process is distributed as a negative binomial process, i.e, wait for a number of successes and requires a different statistical framework to compare models to data.

In the following, the process for constructing distributions and populations is described.